To improve the lower bound on \(E_l\) , we propose the approach similar to those described in [1]}. Namely, we generate nondecreasing PDPs \(\mathcal {D}\)  with \(\mathcal {D}_{\phi } \le d[l,l-\phi ]\)  satisfying Lemmas REF , REF , and which are LP-valid [2]}. We also require that \(\frac{1}{l} \sum _{i=0}^{l-1} \log _l \mathcal {D}_i\)  should be larger than existing lower bounds on \(E_l\) . Then, we run Alg. REF  for all valid \(\mathcal {D}\) . Due to the tremendous search space, for some PDP \(\mathcal {D}\)  the algorithm may not produce a kernel for a long time. In this case, it should be terminated. Nevertheless, during the computer search we observed that if a kernel with given PDP exists, Alg. REF  often returns it quickly (in a few seconds). This is the advantage of depth-first search, which allows us to perform maximization of \(E_l\) .
